1. Inferring the Mutational History of a Tumor Using Multi-state Perfect Phylogeny Mixtures 
Mohammed El-Kebir, Gryte Satas, Layla Oesper, Benjamin J. Raphael 
Cell Systems 
Volume 3, Issue 1, Pages (July 2016)
DOI: /j.cels
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2. Cell Systems 2016 3, 43-53DOI: (10.1016/j.cels.2016.07.004)
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3. Figure 1
Overview
(A) In the multi-state perfect phylogeny problem, we are given a matrix A whose rows are the state vectors of the taxa. We seek to find a tree T that satisfies the infinite alleles assumption and whose leaves correspond to the taxa.
(B) In the perfect phylogeny mixture deconvolution problem (PPMDP), we do not observe the taxa directly. Instead, our measurements F correspond to mixtures of the leaves of a tree T according to unknown proportions U. The goal is to infer T and U from the frequencies F of each character and state in each sample.
(C) Tumors consist of mixtures of distinct cellular populations (clones). Bulk DNA sequencing data of one or more samples of a tumor yields variant allele frequencies (VAFs) of single-nucleotide variants (SNVs) and read-depth ratios and B-allele frequencies (BAFs) from copy number aberrations (CNAs). These measurements are superpositions of the mutations present in the clones in each sample. We show how to derive F from these measurements; solving the PPMDP yields a clonal tree describing the evolution of the clones in the tumor.
Cell Systems 2016 3, 43-53DOI: ( /j.cels )
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4. Figure 2 Overview of SPRUCE
(A) Input are the VAFs of SNVs and the copy numbers and mixing proportions of CNAs, which are derived from read depth and B-allele frequencies.
(B) Combining the input with our multi-state model for the somatic mutational process produces a collection of compatible state trees for each character (Figure S2).
(C) Two character-state tree pairs are compatible if there exists a perfect phylogeny tree that contains both. We construct the pairwise compatibility graph by considering all such pairs.
(D) A maximal clique in the compatibility graph yields a frequency tensor F and collection S of state trees that are an input to the Cladistic-PPMDP.
(E) For each instance, we construct the multi-state ancestry graph GF, which encodes potential ancestral relationships between character-state pairs. We then enumerate all multi-state perfect phylogeny trees with maximum size in this graph, and we compute the corresponding usage matrices U.
Cell Systems 2016 3, 43-53DOI: ( /j.cels )
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5. Figure 3 Simulation Results
(A) Number of solutions (log scale) for instances with n = 5 characters, m∈{2,5,10} samples, and varying sequencing coverages. A coverage of inf corresponds to error-free VAFs.
(B) Solution space of an n = 5, m = 5 instance with error-free VAFs. Vertices are from the solution trees, and each edge is labeled by the number of trees in which it occurs. Red edges indicate the simulated tree.
(C) Recall of SPRUCE’s representative trees is compared to PhyloWGS’s maximum likelihood trees on 60 instances with n = 5 and error-free VAFs (shown in Figure 2A).
(D) Recall values on 20 instances with noisy VAFs (1,000x), n = 5, and m = 10. We run SPRUCE with varying running time limit (N) per state tree combination in seconds.
Cell Systems 2016 3, 43-53DOI: ( /j.cels )
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6. Figure 4
SPRUCE Analysis of Prostate Cancer Sample A22 Reveals Different Phylogenies
(A) The number of characters (SNVs) with distinct CCF intervals in individual samples, computed from the set of compatible state trees of each character. Sizes of circles correspond to the number of characters. PPFIA1 has distinct CCF intervals in four samples.
(B) PFFIA1 is compatible with two state trees, S1 and S2. We show for each sample the CCF interval for S1 (blue intervals), S2 (red intervals), and the cluster CCF (marked × ) reported in Gundem et al. (2015). The Gundem et al. (2015) cluster CCFs are largely consistent with S1.
(C) In sample D (purity ρ=0.772), the 99.9% VAF confidence interval for PFFIA1 and the mixing proportions yield two compatible state trees, S1 and S2, with distinct CCF intervals. Note that S1 adheres to the assumption in Gundem et al. (2015) of a fixed number of mutated copies.
(D) Condensed representation of the resulting perfect phylogeny trees using S1 (left) and S2 (right). Vertices are labeled by the introduced character-state pairs, and colors correspond to clusters reported in Gundem et al. (2015). In the left tree the blue cluster is a child of PFFIA1, whereas in the right tree it is a sibling of PFFIA1. See Figure S5 for the complete trees.
Cell Systems 2016 3, 43-53DOI: ( /j.cels )
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